A strengthening of McConnel's theorem on permutations over finite fields

TL;DR

This paper extends McConnel's theorem by showing that a small-doubling condition on a subset $D\ subseteq \, o \\mathbb{F}_q^*$ suffices to force any function $f:\\mathbb{F}_q\to\mathbb{F}_q$ with $(f(x)-f(y))/(x-y)\in D$ (for $x\neq y$) to have the form $f(x)=a x^{p^j}+b$ with $a\in\mathbb{F}_q^*$ and $0\le j\le n-1$. The main condition is $|DD^{-1}D^{-1}|\le\frac{q+1}{2}$, and the result connects additive/difference constraints to multiplicative structure via the geometry of directions in the affine plane, leading to a strengthened McConnel-type conclusion. A Plünnecke–Ruzsa-based corollary broadens the applicability to sets with small doubling, and the paper discusses implications for the structure of direction sets $\\mathcal{D}_f$ of linearized polynomials, including when such sets are cosets of subgroups of $\mathbb{F}_q^*$. Overall, the work blends finite geometry, additive combinatorics, and the theory of linearized polynomials to generalize a classic permutation-result over finite fields.

Abstract

Let $p$ be a prime, $q=p^n$, and $D \subset \mathbb{F}_q^*$. A celebrated result of McConnel states that if $D$ is a proper subgroup of $\mathbb{F}_q^*$, and $f:\mathbb{F}_q \to \mathbb{F}_q$ is a function such that $(f(x)-f(y))/(x-y) \in D$ whenever $x \neq y$, then $f(x)$ necessarily has the form $ax^{p^j}+b$. In this notes, we give a sufficient condition on $D$ to obtain the same conclusion on $f$. In particular, we show that McConnel's theorem extends if $D$ has small doubling.

Theorems & Definitions (5)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• proof : Proof of Theorem \ref{['thm:main']}
• proof : Proof of Corollary \ref{['cor']}